Problems

Let G be a graph with maximum degree at most three. Suppose that G has an m-covering by 5-cycles, for some positive integer m. Then G is one of the seven graphs depicted in Figure 7.11.

Does the list of 5-capacities, presented in Section 4, include all members of GI_5 ?

Find an efficient algorithm to find L_{G+H}(\lambda,k)=max\left{L_{G}(\lambda,\ell)\cdots L_{H}(\lambda-\ell,k-\ell)\text{ for some }\chi(G)\leq \ell\leq k-1\right}.

Let G ≅∪_j=1^2KT_j be a disjoint union of an even number of path-like trees, all of them of the same order, and such that T_j≠P_2 for j=1,2, ..., 2 K. Is G a super edge-magic graph?

Let f(G) be the number of matchings of G. Is f(G)^1/|G|≥ f(H)^1/|H| (5.3) when H fractionally tiles G?

There are conjectured recurrences for T(m, n, k) (see A197654), but so far they are unproved.

For even value of m, it seems that rn(G(4mk+2m; 1,2m) = 2mk^2 + 2m^2k + 5mk + m^2 + m - k/2, if k is odd; 2mk^2 + 2m^2k + 7mk + m^2 + 2m - k - 1/2, if k is even.

Investigating Proposition 17, is there a more convenient expression for the upper bound based only on the Young diagram (see Figures 2 and 4) of the set of CRGs K(a, c) : H ↔_c K(a, c), ∀ H ∈ F(H)?

Considering the graphs H_3(d) leads to the conjecture of e(G^3)≥ 2e(G),for G regular, connected, and diam(G) ≥ 3.

For even m and n > m ≥ 4, Δ(T_n) ≤ n - m + 2 ⇒ R(T_n, W_m) = 2n - 1.

For positive integers k, d and n with k ≥3, find the largest value f_{k,d}(n) such that every connected graph G of maximum degree at most d and of order n contains a k-tree T with |T|≥f_{k,d}(n).

In PG(3, q) and PG(4, q), the upper bounds (1.5), (1.6) hold for all q.

(Covering Radius Conjecture) Let λ ∈ [1/g, g] (recall that g ≥ 1). The covering radius of N_G with respect to the polytope P_{1,λ} is at least √(g/λ)/n where n is the number of vertices of G.

A set S of patterns is uncovered if and only if it satisfies f_n = O(t_n+1).

If r is odd, a basis of MR^sharp will be parametrized by colored compositions such that parts of color 0 are not ≡ 0 bmod r and parts of color 1 are arbitrary. The Hilbert series is then H_r(t)=frac1-t^r1-2(t+t^2+… +t^r). If r is even,…

If G is a critical graph (not banned) and ρ^⊥(G)=d , then G-M(G) (with M a match maximal) has ρ^⊥(G-M(G))≤ d-1

Perhaps an equally daring conjecture would be that L(G) = {d : d divides |G|}, in which case we would have f(G) = f^{*}(|G|).

Let Δ be a Nim-regular complex, F a nonempty face, D_i a minimal cover of F by circuits and D = uplus_i D_i. Is it necessarily true that D - F is not a DUOC?

How can we read the combinatorics of the point configuration S ⊂ TT^X from the split system of C_S ?

Can one prove better upper or lower bounds for ℓ_n ?